Beyond Gaussian approximation: Bootstrap for maxima of sums of independent random vectors
By Hang Deng, Cun-Hui Zhang. Annals of Statistics, 2017
53 citations recorded by the source corpus.
The Bonferroni adjustment, or the union bound, is commonly used to study rate optimality properties of statistical methods in high-dimensional problems. However, in practice, the Bonferroni adjustment is overly conservative. The extreme value theory has been proven to provide more accurate multiplicity adjustments in a number of settings, but only on ad hoc basis. Recently, Gaussian approximation has been used to justify bootstrap adjustments in large scale simultaneous inference in some general settings when $n \gg (\log p)^7$, where $p$ is the multiplicity of the inference problem and $n$ is the sample size. The thrust of this theory is the validity of the Gaussian approximation for maxima of sums of independent random vectors in high-dimension. In this paper, we reduce the sample size requirement to $n \gg (\log p)^5$ for the consistency of the empirical bootstrap and the multiplier/wild bootstrap in the Kolmogorov-Smirnov distance, possibly in the regime where the Gaussian approximation is not available. New comparison and anti-concentration theorems, which are of considerable interest in and of themselves, are developed as existing ones interweaved with Gaussian approximation are no longer applicable.
Subfields: Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/ebd19c54b228d2ec8439763e028463e5e930d5a3
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Citations in the corpus
- Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, methodology): “However, since we only have upper boundson the approximation error, this does not imply that the multiplier bootstrap is necessarily more precise than the empirical bootstrap. In fact, simulations in [20, 17, 19] suggest the opposite may be true, with approximation errors being rather similar for mo”
- Improved central limit theorem and bootstrap approximations in high dimensions (2019, unknown): “This theorem improves upon the bounds in [20], who obtained a similar result with the rate 1/6 instead of 1/4.”
- High-dimensional econometrics and regularized GMM (2018, background): “[58] show that Condition E, which is imposed in Theorems 2.”